Bispectral Operator Algebras

Abstract

This dissertation is an amalgamation of various results on the structure of bispectral differential operator algebras, ie. algebras of differential operators with possibly noncommutative coefficients in a variable $x$ satisfying the property of having a family $\psi(x,y)$ of joint eigenfunctions which are also eigenfunctions of another operator in the \emph{spectral} parameter $y$. In this docume nt, we extend the modern theory of commuting differential operators to differential operators with noncommutative coefficients. We prove under fairly general circumstances that such algebras are isomorphic to endomorphism rings of torsion- free modules on rational curves. We also classify all rank $1$ noncommutative bispectral differential operator algebras and explore the role of Darboux transformations in the construction of bispectral differential operator algebras, particularly for the bispectral operator algebras associated to a weight matrix $w$.